C-group


In mathematical group theory, a C-group is a group such that the centralizer of any involution has a normal Sylow 2-subgroup. They include as special cases CIT-groups where the centralizer of any involution is a 2-group, and TI-groups where any Sylow 2-subgroups have trivial intersection.
The simple C-groups were determined by, and his classification is summarized by. The classification of C-groups was used in Thompson's classification of N-groups.
The finite non-abelian simple C-groups are
  • the projective special linear groups PSL2 for p a Fermat or Mersenne prime, and p≥5
  • the projective special linear groups PSL2
  • the projective special linear groups PSL2 for n≥2
  • the projective special linear groups PSL3 for n≥1
  • the projective special unitary groups PSU3 for n≥2
  • the Suzuki groups Sz for n≥1

    CIT-groups

The C-groups include as special cases the CIT-groups, that are groups in which the centralizer of any involution is a 2-group. These were classified by, and the finite non-abelian simple ones consist of the finite non-abelian simple C-groups other than PSL3 and PSU3 for n≥2. The ones whose Sylow 2-subgroups are elementary abelian were classified in a paper of, which was forgotten for many years until rediscovered by Feit in 1970.

TI-groups

The C-groups include as special cases the TI-groups, that are groups in which any two Sylow 2-subgroups have trivial intersection. These were classified by, and the simple ones are of the form PSL2, PSU3, Sz for q a power of 2.